## Prodromi catoptricorum et dioptricorum, sive conicorum operis ad abdita radii reflexi et refracti mysteria praevii & facem praeferentis. Libri quatuor priores [Liber primus et secundus]. D. A. L. G.

Paris: Jean Dedin, 1631-39.

First edition, extremely rare author’s presentation copy, of all four books of this important work on conic sections, intended to provide the geometrical basis for the study of optics. “Mydorge’s work on conic sections contains hundreds of problems published for the first time, as well as a multitude of ingenious and original methods that later geometers frequently used, usually without citing their source” (DSB). Books I and II (pp. 1-134) were first published separately in 1631; a second edition appeared in 1639 with two additional books. The present copy has the first edition of the first two books, with the 1631 title page, bound up with the last two books from the second edition. A printed paper slip *Libri quatuor priores* has been pasted over *Liber primus et secundus* on the title to accommodate the added books, and a large section of text has also been pasted over the original on page 67 corresponding to changes in book I made between the 1631 and 1639 editions. The 1631 edition is very much rarer than the 1639: OCLC lists only five copies of the former – Danish Royal Library (but this copy is actually of the 1641 edition), Columbia, NYPL, Zürich, BNF – but 24 of the latter; COPAC adds copies of the 1631 edition at Oxford, Cambridge and UCL. It is likely that the 1631 edition was printed in very small numbers and was mostly, if not entirely, intended for presentation: the copies at Columbia, Zürich and BNF all have authorial corrections. The only other copy of the 1631 edition to have appeared at auction was Michel Chasles’ copy, last sold in 1972.

“In his study of conic sections Mydorge continued the work of Apollonius, whose methods of proof he refined and simplified ... Mydorge asserts that if from a given point in the plane of a conic section radii to the points of the curve are drawn and extended in a given relationship, then their extremities will be on a new conic section similar to the first. This statement constitutes the beginnings of an extremely fruitful method of deforming figures; it was successfully used by La Hire and Newton, and later by Poncelet and, especially, by Chasles, who named it deformation homographique.

“Mydorge posed and solved the following problem in [book] III: ‘On a given cone place a given conic section’ – a problem that Apollonius had solved only for a right cone. Mydorge was also interested in geometric methods used in approximate construction, such as that of a regular heptagon. Another problem that Mydorge solved by approximation – although he did not clearly indicate his method – was that of transforming a square into an equivalent regular polygon possessing an arbitrary number of sides” (DSB)

“Mydorge (1585-1647) was born in Paris to one of the wealthiest families in France. He was educated at the Jesuit College of La Flèche and subsequently trained as a lawyer, before embarking on a legal and administrative career. After serving as conseiller to the court of the Grand Châtelet, he became treasurer of the généralité of Amiens, the collector general being a direct agent of the king. Mydorge's chosen employment allowed him sufficient time to combine public office with the life of a savant. Residing in what remained of the ancient Palais des Tournelles, he first met Descartes around 1625, becoming one of his most faithful friends and helping to establish his reputation in Paris. The mathematician Claude Hardy, a leading figure in the scientific circles around Mersenne, Roberval, and Étienne Pascal, lodged with him while he was producing his edition of Euclid's Elements.

“Mydorge shared with Descartes a strong interest in optics and the nature of vision. It is well known that in order to promote his friend's investigations on these topics, he commissioned the production of innumerable parabolic, hyperbolic, oval, and elliptic lenses, reputedly spending in excess of 100,000 écus on optical instruments over the years. Both men were interested particularly in refraction, and when Descartes, independently of Snell, discovered the law of refraction, he persuaded Mydorge to have a hyperbolic glass made in order to test his discovery” (Cambridge Descartes Lexicon). Mydorge was also a friend of Fermat and Mersenne, and in 1638 played a role in settling the dispute between Descartes and Fermat that had arisen when Fermat refuted Descartes’ *Dioptrique.*

Mydorge played an important role in Descartes’ discovery of the sine law of refraction, which the two men almost certainly formulated in 1626 (Sasaki, p. 175). “In a letter to Mersenne from around 1627, Mydorge used a rule to calculate angles of refraction, given the angles of one pair of incident and refracted rays … The rule comes down to a cosecant ‘law’ … Later in the letter, Mydorge applied this rule to lenses and transformed it into sine form. Mydorge’s rule embodies thetwo assumptions that formed thecore of Descartes’ derivation ofthe sine law in *La Dioptrique*” (pp. 126-127).

Publication of the present work was sponsored by Sir Charles Cavendish (?1595-1654), to whom the book is dedicated – Mydorge calls Cavendish “extremely skilled in all mathematics and a very dear friend to me” (Malcolm & Stedall, p. 88). Cavendish seems to have developed contacts with foreign mathematicians and by the summer of 1631 was corresponding with Mydorge. In 1671 John Collins wrote “'they complaine in france (as we doe here) that their Booksellers will not undertake to print mathematicall Bookes there, thence it came to passe that the four latter books of Mydorge were never printed, as the former had not been unless Sir Charles Cavendish had given 50 crownes as a Dowry with it” (Collins to Gregory, 14 March 1671/2 in Newton, *Correspondence* 47). The manuscript of ‘the four latter books’ was apparently taken to England by Cavendish’s brother William and Thomas Wriothesley, Earl of Southampton, and then lost. The first four books were reissued in 1641 and 1660, and under the title *De sectionibus conicis* were included by Mersenne in his *Universae geometriae* (1644).

Albert *et al*. 1636; B. O. A. C. P. 72; Macclesfield 1507 (second edition, lacking four pages of preliminaries and the errata leaf). Malcolm & Stedall, John Pell (1611-1685) and His Correspondence with Sir Charles Cavendish, 2005. Sasaki, *Descartes’ Mathematical Thought*, S2003 (pp. 172-5).

Folio (134 x 228 mm), pp. [x], 308, woodcut printer’s device on title and numerous woodcut diagrams in text, ‘Dono authoris’ written in a contemporary hand at top of title page, another inscription dated 1833 on title, without the two errata leaves as often. Contemporary reversed calf (worn), red lettering piece on spine (some browning as usual).

Item #3576

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Price:
$13,500.00
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